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TMUA 2018 · Paper 1 · Question 12 of 20

TMUA 2018 Paper 1 Question 12

Differentiation and integration — Areas versus signed integrals. Try it first; the answer and a full worked solution are below.

TMUA 2018 · Paper 1Differentiation and integrationAreas versus signed integrals6 options
A curve has equation y = f(x), where f(x) = x(x − p)(x − q)(r − x) with 0 < p < q < r.

You are given that:

∫0r f(x) dx = 0 ∫0q f(x) dx = −2 ∫pr f(x) dx = −3

What is the total area enclosed by the curve and the x-axis for 0 ≤ x ≤ r?

  1. A0
  2. B1
  3. C4
  4. D5
  5. E6
  6. F10
Show the answer and worked solution
answer · F
  1. A0
  2. B1
  3. C4
  4. D5
  5. E6
  6. F10
The roots are 0, p, q, r, and checking signs shows the curve is above the axis on (0,  p), below on (p,  q) and above again on (q,  r). Write the three areas as A, B, C, so the signed integrals over those intervals are A, −B, C. The given data become A − B + C = 0, A − B = −2 and −B + C = −3. The first two give C = 2, then the third gives B = 5 and hence A = 3. Total area is A + B + C = 3 + 5 + 2 = 10; the value 0 is the trap of adding the signed pieces instead.